Friday, 2 May 2025

A Variation on the Descent to Zero

In my post on the 23rd of April 2025 titled Descent to Zero, I considered the smallest numbers that take a certain number of steps to reach 0 under "k max product of two numbers whose concatenation is k". What happens if we change this slightly so that the rule is now "k max product of two prime numbers whose concatenation is k".

This is highly restrictive because only numbers that can be split into a pair of prime numbers in one or more ways are eligible for consideration. For example, 246 is dismissed but 235 is eligible for consideration because it can be split into 23 x 5. Moreover, the process of splitting into primes needs to continue until zero is reached if the number is to be a candidate for the smallest number. Here are the numbers that require from 1 to 9 steps to reach zero:1,22,55,115,235,475,3389,13457,35743The breakdown is as follows:

  • Descent of 9 steps to zero: 35743 --> 17229, 3893, 1167, 737, 511, 55, 25, 10, 0
  • Descent of 8 steps to zero: 13457 --> 5941, 4705, 235, 115, 55, 25, 10, 0
  • Descent of 7 steps to zero: 3389 --> 1167, 737, 511, 55, 25, 10, 0
  • Descent of 6 steps to zero: 475 --> 235, 115, 55, 25, 10, 0
  • Descent of 5 steps to zero: 235 --> 115, 55, 25, 10, 0
  • Descent of 4 steps to zero: 115 --> 55, 25, 10, 0
  • Descent of 3 steps to zero: 55 --> 25, 10, 0
  • Descent of 2 steps to zero: 22 --> 4, 0
  • Descent of 1 step to zero: 1 --> 0
The convention is that single digits or 10 get mapped to zero. Let's look at how 35743 reaches zero:357433×5743=172291722917×229=574357435×743=38933893389×3=1167116711×67=73773773×7=5115115×11=55555×5=25252×5=10100Note that other prime number products are possible. For example 5743 could be split into 57 x 43 but this is smaller than 5 x 5743. Similarly, 737 could be split into 7 x 37 but again this is smaller than 73 x 7.

There are in fact only 127 numbers in the range from 11 to 40000 that can be reduced down to 10 or a single digit. They are (permalink) with record breakers shown in red:

22, 23, 25, 32, 33, 52, 55, 112, 113, 115, 202, 203, 205, 211, 235, 297, 302, 303, 311, 415, 475, 502, 505, 511, 523, 541, 547, 583, 729, 737, 773, 835, 1012, 1013, 1015, 1102, 1103, 1105, 1153, 1167, 1512, 1675, 2002, 2003, 2005, 2011, 2101, 2151, 2251, 2305, 2512, 3002, 3003, 3011, 3101, 3389, 3893, 4015, 4105, 4437, 4615, 4705, 5002, 5005, 5011, 5023, 5041, 5047, 5083, 5101, 5167, 5401, 5461, 5821, 5941, 6711, 7029, 7073, 7171, 7443, 8215, 8305, 9415, 10102, 10103, 10105, 10171, 11002, 11003, 11005, 11053, 11067, 11191, 11491, 11643, 12743, 13457, 14537, 15102, 16705, 17229, 19111, 20002, 20003, 20005, 20011, 20101, 20151, 20251, 23005, 24183, 25051, 25102, 25501, 29659, 30002, 30003, 30011, 30101, 30389, 31971, 32237, 33881, 35743, 36437, 38813, 38903

This permalink will allow you to enter any of the above numbers and receive as output the descent of the number to 10 or a single digit. For example, entering the number 38903 produces the following output:
Starting with: 38903
Dividing 38903 into prime parts and multiplying gives: 1167
Dividing 1167 into prime parts and multiplying gives: 737
Dividing 737 into prime parts and multiplying gives: 511
Dividing 511 into prime parts and multiplying gives: 55
Dividing 55 into prime parts and multiplying gives: 25
Dividing 25 into prime parts and multiplying gives: 10
Reached: 10

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